One can read in the Wikipedia page for "Gödel's incompleteness theorems":
Undecidability of a statement in a particular deductive system does
not, in and of itself, address the question of whether the truth value
of the statement is well-defined, or whether it can be determined by
other means. Undecidability only implies that the particular deductive
system being considered does not prove the truth or falsity of the
statement. Whether there exist so-called "absolutely undecidable"
statements, whose truth value can never be known or is ill-specified,
is a controversial point in the philosophy of mathematics.
NB: The same text appears in the Wikipedia page for "Undecidable Problem".
I don't understand this. It seems to me that there are a couple of theorems in mathematical logic which, on the contrary, very clearly explain the relation between the undecidability of a statement and its "truth value": depending on the meaning of "truth value", I am thinking about Post's tautology theorem and Gödel's completeness theorem.
Am I missing something? And what does Wikipedia mean by "absolutely undecidable"?
Let me elaborate in a little bit for clarity. My understanding is that by the completeness theorem, a statement is undecidable if and only if there exist models in which it is true and other models in which it is false. Moreover (or alternatively), by the tautology theorem of Post, a statement is undecidable if and only if there exist some truth valuations for which it is true and others for which it is false. In any case, the conclusion, it seems to me, is simply that a statement is undecidable if and only if its truth value is not defined (it can be "chosen" true or false arbitrarily).
EDIT. Let me add a couple of observations after reading the answers of 6005, user21820, and spaceisdarkgreen, which are not entirely satisfactory to me. These answers are defending Wikipedia's text by interpreting the meaning of "truth value" relatively to some kind of correct model or worse, the physical world. Neither of these notions have a place in mathematical logic, it seems to me. When talking about natural numbers, we may like to think there is a correct model, but it would be silly to assume that there is a "preferred universe" for every theory.
For example, take Euclid's 5 axioms for geometry, remove axiom #5 (the "parallel postulate") so that you're left with only the first 4 axioms (you get "absolute geometry"). Both the Euclidean plane and the hyperbolic plane are models for this theory. Is one of the two the "correct model"? Clearly no, since we got rid of the fifth axiom that would discriminate between the two.
So at this point I still find that Wikipedia's assertion about "truth value" is still irrelevant.
Best Answer
It is true that if a statement is undecidable (i.e., not provable one way or the other, in a given deductive system), then there are models in which it is true and models in which it is false. This is not, however, the only way to interpret "whether the truth value of the statement is well-defined". In the case of $\mathbb{N}$, in particular, it is common to believe that there are the "actual" natural numbers $0, 1, 2, 3, \ldots$, and all other models of arithmetic are fake, or nonstandard models. In this sense, we believe the following: every statement about the natural numbers is either true or false. (This is true even in mathematical logic, where for example we talk about properties of the "standard" and "nonstandard" models of $\mathbb{N}$.)
Godel's theorem, however, can be interpreted as calling into question this very assertion! After all, if we can never precisely describe the set of natural numbers $\mathbb{N}$ -- not in PA, nor in ZFC, nor in any other (recursively axiomatizable) theory -- then is there really only one $\mathbb{N}$ that exists? Does it really make sense to say that every statement about the natural numbers is either true or false?
Some would say yes, and some would say no. It's a philosophical matter, because the question is whether you believe that there is an ideal mathematical universe out there beyond what we can ever formalize. That is the controversy that Wikipedia is talking about in this paragraph.